Research

I work in Number Theory. More specifically, my main interst lies in Arithmetic Geometry, where I work on Elliptic curves, Drinfeld modules, Drinfeld modular forms and related topics. Other interests include Ducci sequences, orders of elements in finte fields and most recently quadratic number fields and the Cohen-Lenstra-Martinet heuristics.

Publications

Note: for copyright reasons, some of the following PDF files are not the final, published versions. Also, this is page is under contruction, so most links are still broken.
  1. Drinfeld modular polynomials of level T (with Mahefason Heriniaina Razafinjatovo). [ arXiv:2412.11324] [GitHub]

  2. Heights of Drinfeld modular polynomials and Hecke images (with Fabien Pazuki and Zhenlin Ran). [ arXiv:2410.11132]

  3. Parity bias in fundamental units of real quadratic fields. (with Cameron Shaw-Carmody), Bull. Aust. Math. Soc., to appear. [PDF] [GitHub]

  4. SeCritMass: Threshold Secret Petitions. arXiv:2408.01255 [cs.CR]

  5. Explicit bounds on the coefficients of modular polynomials and the size of X0(N). (with Fabien Pazuki and Desirée Gijón Gómez) Proc. London Math. Soc., to appear. [PDF] [GitHub]

  6. Explicit bounds on the coefficients of modular polynomials for the elliptic j-invariant. (with Fabien Pazuki) Proc. Amer. Math. Soc. Ser. B 11 (2024), 277–286. [PDF]

  7. Drinfeld modular forms of arbitrary rank (With Dirk Basson and Richard Pink). [PDF]. Mem. Amer. Math. Soc. 304 (2024) no. 1531, 94 pages. [Published version].
    This monograph updates (e.g. Prop 5.16) and combines the three preprints:
  8. Multiplicative orders of Gauss periods and the arithmetic of real quadratic fields. Finite Fields Appl., 73 (2021). [PDF] arXiv:2006.10344

  9. Generic monodromy of Drinfeld modular varieties in special characteristic. (With Gebhard Böckle) In: "Abelian Varieties and Number Theory" (M. Jarden, T. Shaska, eds.) Contemp. Math., Amer. Math. Soc. 767, (2021), 147--160. [PDF] arXiv:1912.09820

  10. Heights and isogenies of Drinfeld modules. (With Fabien Pazuki and Heriniaina Razafinjatovo) Acta Arith. 197 no.2 (2021), 111-128. arXiv:1908.03485 [PDF]

  11. Lower bounds for periods of Ducci sequences. (With Igor Shparlinski) Bull. Aust. Math. Soc. 102 no.1 (2020), 31-38. [PDF] arXiv:1909.04462

  12. Periods of Ducci sequences and odd solutions to a Pellian equation. Bull. Aust. Math. Soc. 100 no.2 (2019), 201-205. [PDF]

  13. On certain Drinfeld modular forms of higher rank. (with Dirk Basson). J. Théor. Nombres Bordeaux 29 no.3 (2017), 827-843. [PDF]

  14. A note on Gekeler's h-function. Arch. Math. 107 (2016), 305-313. [PDF] The final publication is available at Springer via http://dx.doi.org/doi:10.1007/s00013-016-0923-1. arXiv:1605.02567

  15. Drinfeld modular polynomials in higher rank II: Kronecker congruences. (With Hans-Georg Rück). J. Number Theory 165 (2016), 1-14. [PDF] arXiv:1303.4086

  16. Explicit Drinfeld moduli schemes and Abhyankar's generalized iteration conjecture. J. Number Theory 160 (2016), 432-450. [PDF] arXiv:1503.06420

  17. Newton identities for Weierstrass products. Amer. Math. Monthly 119, no. 9 (2012), 796-799. [PDF]

  18. Special subvarieties of Drinfeld modular varieties. J. reine angew. Math. 668 (2012), 35-57. [PDF] arXiv:math/0503452

  19. Ducci sequences and cyclotomic fields. J. Difference Equ. Appl. 16 (2010), no. 7, 847-862. [PDF]

  20. Torsion bounds for elliptic curves and Drinfeld modules. J. Number Theory 130 (2010), 1241-1250. [PDF]

  21. Drinfeld modular polynomials in higher rank. J. Number Theory 129 (2009), 59-83. (With Hans-Georg Rück) [PDF] arXiv:math/0701291

  22. Heegner points and the rank of elliptic curves over large extensions of global fields. Canadian J. Math. 60 (2008), no.3, 481-490. (With Bo-Hae Im) [PDF] arXiv:math/0604107

  23. The André-Oort conjecture for Drinfeld modular varieties. C. R. Math. Acad. Sci. Paris, Ser. I, 344 (2007), 733-736. [PDF]

  24. Ducci sequences in higher dimensions. Integers: Electronic Journal of Combinatorial Number Theory 7 (2007), #A24. [PDF]

  25. Ducci sequences and cyclotomic polynomials. Finite Fields Appl. 13 (2007), 293-304. (With Ernest Lötter and Brink van der Merwe.) [PDF]

  26. CM points on products of Drinfeld modular curves. Trans. Amer. Math. Soc. 359 (2007), 1351-1374. [PDF] arXiv:math/0407334

  27. Monodromy groups associated to non-isotrivial Drinfeld modules in generic characteristic. In: "Number Fields and Function Fields - Two Parallel Worlds", (G. van der Geer, B. Moonen, R. Schoof, eds.), Progress in Mathematics, 239, p61-69, Birkhäuser Verlag, 2005. (With Richard Pink.) [PDF] arXiv:math/0407335

  28. The André-Oort conjecture for products of Drinfeld modular curves. J. reine angew. Math. 579 (2005), 115-144. [PDF] arXiv:math/0303038

  29. Images of isogeny classes on modular elliptic curves. Math. Res. Lett. 11 (2004), 649-651. [PDF] arXiv:math/0407336

  30. Higher Heegner points on elliptic curves over function fields. J. Number Theory 104 (2004), 315-326. [PDF] arXiv:math/0304216

  31. La conjecture d'André-Oort pour le produit de deux coubres modulaires de Drinfeld. C. R. Math. Acad. Sci. Paris 335 (2002), no. 11, 867--870. [PDF]

  32. Distinguished liftings and the André-Oort conjecture. Quaest. Math. 25(3) (2002), 363-380. [PDF]

  33. Heights of CM points on complex affine curves. Ramanujan J. 5 (2001), 311-317. [PDF]

  34. Ducci sequences over abelian groups. Comm. Algebra 27.12 (1999), 5999-6013. [PDF]

  35. A note on a paper by Glaser and Schöffl. Fibonacci Quart. 36.5 (1998), 463-466. [PDF]

  36. Strategy and complexity of the game of squares. Bull. London. Math. Soc 30 (1998), 274-282. (With Mike Robson) [PDF]

  37. Line profile variations in the spectra of the gamma Dor star HR 2740, In: IAU Symposium 185 "New Eyes to See Inside the Sun and Stars", (F.-L. Deubner, J. Christensen-Dalsgaard, D. Kurtz, eds) Kyoto, Japan, 18-22 August, 1997, p. 387. (5th author, with E. Poretti, L. Mantegazza, C. Koen, P. Martinez, D. De Alwis and H. Haupt). [PDF]

  38. Discovery and analysis of gamma doradus type pulsations in the F0 IV star HR 2740 QW pup, Mon. Not. R. Astron. Soc. 292 (1997), 621-630. (4th author, with E. Poretti, C. Koen, P. Martinez, D. de Alwis and H. Haupt) [PDF]